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录入26届高一寒假作业6并建立related

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题库0.3/Problems.json
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@ -87855,7 +87855,9 @@
"20220701\t王伟叶"
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@ -88867,7 +88869,9 @@
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@ -161186,7 +161190,9 @@
"20220720\t王伟叶"
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@ -434447,7 +434453,9 @@
"20230503\t王伟叶"
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@ -546144,7 +546152,9 @@
"20221231\t王伟叶"
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@ -548407,7 +548417,9 @@
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@ -627020,6 +627032,397 @@
"space": "4em",
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},
"023463": {
"id": "023463",
"content": "若函数 $y=x^2-x+3$ 的定义域为 $(0,+\\infty)$, 则它的值域是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
"edit": [
"20240108\t王伟叶"
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"same": [],
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},
"023464": {
"id": "023464",
"content": "函数 $y=3^x+\\ln (1+x)$ 的近似零点为\\blank{50}(精确到$0.01$).",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
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"023465": {
"id": "023465",
"content": "$f(x)=x^2+x+1$, $g(x)=x^2+1$, 则 $y=2 g(x)-f(x)$ 的最小值是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"023466": {
"id": "023466",
"content": "若函数 $y=x^2-3 x-4$ 的定义域为 $[0, m]$, 值域为 $[-\\dfrac{25}{4},-4]$, 则 $m$ 的取值范围是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"same": [],
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"space": "",
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"023467": {
"id": "023467",
"content": "若函数 $f(x)$ 的定义域是 $[0,1]$, 则 $f(x+a)+f(x-a)$($0<a<\\dfrac{1}{2}$) 的定义域是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"related": [
"005334"
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"remark": "",
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"023468": {
"id": "023468",
"content": "下列四组函数中, 表示同一函数的组别有\\blank{50}.\\\\\n\\textcircled{1} $f(x)=x$, $g(x)=(\\sqrt[2 n]{x})^{2 n}$($n \\in \\mathbf{N}$, $n \\geq 1$);\\\\\n\\textcircled{2} $f(x)=x$, $g(x)=\\sqrt[2 n+1]{x^{2 n+1}}$($n \\in \\mathbf{N}$, $n \\geq 1$);\\\\\n\\textcircled{3} $f(n)=2 n-1(n \\in Z)$, $g(n)=2 n+1$($n \\in Z$);\\\\\n\\textcircled{4} $f(x)=x^2-2 x-1$, $g(t)=t^2-2 t-1$.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"edit": [
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"same": [],
"related": [
"002823"
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"remark": "",
"space": "",
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},
"023469": {
"id": "023469",
"content": "下列命题: \\textcircled{1} 偶函数的图像一定与 $y$ 轴相交; \\textcircled{2} 奇函数的图像一定通过原点; \\textcircled{3} 不存在既是奇函数又是偶函数的函数; \\textcircled{4} 偶函数的图像关于 $y$ 轴对称. 其中正确的有\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
"edit": [
"20240108\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"023470": {
"id": "023470",
"content": "$f(x)$ 是 $(-\\infty, +\\infty)$ 上的奇函数, $f(x+2)=-f(x)$, 当 $0 \\leq x \\leq 1$ 时, $f(x)=x$, 则 $f(7.5)=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
"edit": [
"20240108\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
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},
"023471": {
"id": "023471",
"content": "对定义域为 $\\mathbf{R}$ 的任何奇函数 $f(x)$, 都有\\bracket{20}.\n\\fourch{$f(x)-f(-x)>0$}{$f(x)-f(-x) \\leq 0$}{$f(x) \\cdot f(-x) \\leq 0$}{$f(x) \\cdot f(-x)>0$}",
"objs": [],
"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
"edit": [
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],
"same": [],
"related": [],
"remark": "",
"space": "",
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},
"023472": {
"id": "023472",
"content": "已知函数 $f(x), g(x)$ 定义在同一区间 $D$ 上, $f(x)$ 是增函数, $g(x)$ 是减函数, 且 $g(x) \\neq 0$, 则在 $D$ 上\\bracket{20}.\n\\twoch{$f(x)+g(x)$ 一定是减函数}{$f(x)-g(x)$ 一定是增函数}{$f(x) \\cdot g(x)$ 一定是增函数}{$\\dfrac{f(x)}{g(x)}$ 一定是减函数}",
"objs": [],
"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"023473": {
"id": "023473",
"content": "某种奥运纪念品进货价 50 元/件, 据市场调查, 当销售价格 $x$ (元/件) 在 $x \\in[50,80]$ 时,每天售出件数 $p=\\dfrac{100000}{x-40}$, 若想每天获得的利润最大, 销售价格应定为多少元?",
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"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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],
"same": [],
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"remark": "",
"space": "4em",
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"023474": {
"id": "023474",
"content": "设 $x_1, x_2$ 是关于 $x$ 的一元二次方程 $x^2-2(m-1) x+m+1=0$ 的两个实根, 又 $y=x_1^2+x_2^2$.\\\\\n(1) 求 $y=f(m)$ 的解析式及此时函数的定义域;\\\\\n(2) 指出 $f(m)$ 的单调区间.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"same": [],
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"023475": {
"id": "023475",
"content": "设函数 $f(x)$ 是定义在 $\\mathrm{R}$ 上的偶函数, 并在区间 $(-\\infty, 0)$ 上是严格增函数, $f(2 a^2+a+1)<f(3 a^2-2 a+1)$, 试确定实数 $a$ 的取值范围.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
"edit": [
"20240108\t王伟叶"
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"same": [],
"related": [
"002787",
"015785"
],
"remark": "",
"space": "4em",
"unrelated": []
},
"023476": {
"id": "023476",
"content": "$f(x)=x^2+x+1$, $g(x)=x^2+1$, 则 $\\dfrac{f(x)}{g(x)}$ 的最大值是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"same": [],
"related": [
"020630"
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"remark": "",
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"023477": {
"id": "023477",
"content": "若函数 $f(x)=a|x-b|+2$ 在 $[0,+\\infty)$ 上为严格增函数, 则实数 $a, b$ 的取值范围是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
"edit": [
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],
"same": [],
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"remark": "",
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"023478": {
"id": "023478",
"content": "已知 $a$ 是常数, 若函数 $y=|x-a|+3-x$ 的函数值恒为非负, 则 $a$ 的取值范围为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
"edit": [
"20240108\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "",
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},
"023479": {
"id": "023479",
"content": "已知函数 $f(x)$ 对任意 $x, y \\in \\mathbf{R}$ 都有 $f(x+y)=f(x)+f(y)$, 且当 $x>0$ 时, $f(x)<0$, $f(1)=-2$.\\\\\n(1) 判断函数 $f(x)$ 的奇偶性;\\\\\n(2) 当 $x \\in[-3,3]$ 时, 函数 $f(x)$ 是否有最值?如果有, 求出最值; 如果没有, 请说明理由.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
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"20240108\t王伟叶"
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"related": [
"020543"
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"space": "4em",
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"023480": {
"id": "023480",
"content": "设 $[x]$ 表示不大于 $x$ 的最大整数, 解方程 $2 x-[x]=5$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"edit": [
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"same": [],
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"space": "4em",
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"023481": {
"id": "023481",
"content": "对于 $y=f(x)$ 是定义在 $(0,+\\infty)$ 上的函数, 若其满足: 对任意 $x>0$, 都成立 $f(x)=f(\\dfrac{1}{x})$, 就称 $y=f(x)$ 是``倒数对称''的.\\\\\n(1) 判断函数 $y=x-\\dfrac{1}{x}$ 与函数 $y=x^2+\\dfrac{1}{x^2}$ 是否是``倒数对称''的;\\\\\n(2) 若``倒数对称''的函数 $y=f(x)$ 在区间 $[1,2]$ 上是严格减函数, 判断它在区间 $[\\dfrac{1}{2}, 1]$ 上的单调性, 并说明理由;\\\\\n(3) 证明: 若 $y=f(x)$ 是``倒数对称''的函数, 则存在定义在 $[2,+\\infty)$ 上的函数 $y=g(x)$,使得对任意 $x \\in (0,+\\infty)$, 总成立 $f(x)=g(x+\\dfrac{1}{x})$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"space": "4em",
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"030001": {
"id": "030001",
"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",